Logspace Computations in Graph Groups and Coxeter Groups
نویسندگان
چکیده
Computing normal forms in groups (or monoids) is in general harder than solving the word problem (equality testing). However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal forms for important classes of groups. We show that shortlex normal forms in graph groups and in right-angled Coxeter groups can be computed in logspace. Graph groups are also known as free partially commutative groups or as right-angled Artin groups in the literature. (Artin groups can be realized as subgroups of Coxeter groups.) Graph groups arise in many areas and have a close connection to concurrency theory. The connection is used here. Indeed, for our result we use a representation of group elements by Mazurkiewicz traces. These are directed acyclic nodelabelled graphs (i.e. pomsets). They form an algebraic model to describe runs of concurrent systems. Concurrent systems which are deterministic and co-deterministic can be studied via inverse monoids. As an application of our results we show that the word problem for free partially commutative inverse monoids is in logspace. This result generalizes a result of Ondrusch and the third author on free inverse monoids. All Coxeter groups are linear, so the word problem can be solved in logspace, but it is open (in the non-right-angled case) whether shortlex normal forms can be computed in logspace, or, less demanding, whether they can be computed efficiently in parallel. We show that for all Coxeter groups the set of letters occurring in the shortlex normal form of an element can be computed in logspace.
منابع مشابه
Logspace computations in Coxeter groups and graph groups
Computing normal forms in groups (or monoids) is computationally harder than solving the word problem (equality testing), in general. However, normal form computation has a much wider range of applications. It is therefore interesting to investigate the complexity of computing normal forms for important classes of groups. For Coxeter groups we show that the following algorithmic tasks can be so...
متن کاملLogspace computations for Garside groups of spindle type
M. Picantin introduced the notion of Garside groups of spindle type, generalizing the 3-strand braid group. We show that, for linear Garside groups of spindle type, a normal form and a solution to the conjugacy problem are logspace computable. For linear Garside groups of spindle type with homogenous presentation we compute a geodesic normal form in logspace.
متن کاملCohomology computations for Artin groups, Bestvina–Brady groups, and graph products
We compute: the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), of Bestvina–Brady groups of type FP, and of graph products of groups, theL-Betti numbers of Bestvina–Brady groups of type FP overQ, and of graph products of groups, the weighted L-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products...
متن کاملRigidity of Right-angled Coxeter Groups
In mathematics, a group is the set of symmetries of an object. Coxeter groups are a broad and natural class of groups that are related to reflectional symmetries. Each Coxeter group is determined by a diagram, called a labeled graph, that encodes algebraic information about the group. In general, two different labeled graphs can give rise to the same group. It is therefore natural to ask: are t...
متن کامل